5.11 Chemical Bonds

The electron states, or “atomic orbitals”, of the elements discussed in section 5.9 form the basis for the “valence bond” description of chemical bonds. This section summarizes some of the basic ideas involved.

5.11.1 Covalent sigma bonds

As pointed out in section 5.9, helium is chemically inert: its outermost, and only, shell can hold two electrons, and it is full. But hydrogen has only one electron, leaving a vacant position for another 1s electron. As discussed earlier in chapter 5.2, two hydrogen atoms are willing to share their electrons. This gives each atom in some sense two electrons in its shell, filling it up. The shared state has lower energy than the two separate atoms, so the H$_2$ molecule stays together. A sketch of the shared 1s electrons was given in figure 5.2.

Fluorine has one vacant spot for an electron in its outer shell just like hydrogen; its outer shell can contain 8 electrons and fluorine has only seven. One of its 2p states, assume it is the horizontal axial state 2p$_z$, has only one electron in it instead of two. Two fluorine atoms can share their unpaired electrons much like hydrogen atoms do and form an F$_2$ molecule. This gives each of the two atoms a filled shell. The fluorine molecular bond is sketched in figure 5.9 (all other electrons have been omitted.)

Figure 5.9: Covalent sigma bond consisting of two 2p$_z$ states.

This bond between p electrons looks quite different from the H$_2$ bond between s electrons in figure 5.2, but it is again a covalent one, in which the electrons are shared. In addition, both bonds are called “sigma” bonds: if you look at either bond from the side, it looks rotationally symmetric, just like an s state. (Sigma is the Greek equivalent of the letter s; it is written as $\sigma$.)

Key Points

Two fluorine or similar atoms can share their unpaired 2p electrons in much the same way that two hydrogen atoms can share their unpaired 2s electrons.

Since such bonds look like s states when seen from the side, they are called sigma or $\sigma$ bonds.

5.11.2 Covalent pi bonds

The N$_2$ nitrogen molecule is another case of covalent bonding. Nitrogen atoms have a total of three unpaired electrons, which can be thought of as one each in the 2p$_x$, 2p$_y$, and 2p$_z$ states. Two nitrogen atoms can share their unpaired 2p$_z$ electrons in a sigma bond the same way that fluorine does, longitudinally.

However, the 2p$_x$ and 2p$_y$ states are normal to the line through the nuclei; these states must be matched up sideways. Figure 5.10 illustrates this for the bond between the two vertical 2p$_x$ states.

Figure 5.10: Covalent pi bond consisting of two 2p$_x$ states.

This covalent bond, and the corresponding one between the 2p$_y$ states, looks like a p state when seen from the side, and it is called a “pi” or $\pi$ bond.

So, the N$_2$ nitrogen molecule is held together by two pi bonds in addition to a sigma bond, making a triple bond. It is a relatively inert molecule.

Key Points

Unpaired p states can match up sideways in what are called pi or $\pi$ bonds.

5.11.3 Polar covalent bonds and hydrogen bonds

Oxygen, located in between fluorine and nitrogen in the periodic table, has two unpaired electrons. It can share these electrons with another oxygen atom to form O$_2$, the molecular oxygen we breath. However, it can instead bind with two hydrogen atoms to form H$_2$O, the water we drink.

In the water molecule, the lone 2p$_z$ electron of oxygen is paired with the 1s electron of one hydrogen atom, as shown in figure 5.11.

Figure 5.11: Covalent sigma bond consisting of a 2p$_z$ and a 1s state.

Similarly, the lone 2p$_y$ electron is paired with the 1s electron of the other hydrogen atom. Both bonds are sigma bonds: they are located on the connecting line between the nuclei. But in this case each bond consists of a 1s and a 2p state, rather than two states of the same type.

Since the $x$ and $y$ axis are orthogonal, the two hydrogen atoms in water should be at a 90 degree angle from each other, relative to the oxygen nucleus. (Without valence bond theory, the most logical guess would surely have been that they would be at opposite sides of the oxygen atom.) The predicted 90 degree angle is in fair approximation to the experimental value of 105 degrees.

The reason that the actual angle is a bit more may be understood from the fact that the oxygen atom has a higher attraction for the shared electrons, or electronegativity, than the hydrogen atoms. It will pull the electrons partly away from the hydrogen atoms, giving itself some negative charge, and the hydrogen atoms a corresponding positive one. The positively charged hydrogen atoms repel each other, increasing their angle a bit. If you go down one place in the periodic table below oxygen, to the larger sulfur atom, H$_2$S has its hydrogen atoms under about 93 degrees, quite close to 90 degrees.

Bonds like the one in water, where the negative electron charge shifts towards the more electronegative atom, are called “polar” covalent bonds.

It has significant consequences for water, since the positively charged hydrogen atoms can electrostatically attract the negatively charged oxygen atoms on other molecules. This has the effect of creating bonds between different molecules called “hydrogen bonds.” While much weaker than typical covalent bonds, they are strong enough to affect the physical properties of water. For example, they are the reason that water is normally a liquid instead of a gas, quite a good idea if you are thirsty, and that ice floats on water instead of sinking to the bottom of the oceans. Hydrogen is particularly efficient at creating such bonds because it does not have any other electrons to shield its nucleus.

Key Points

The geometry of the quantum states reflects in the geometry of the formed molecules.

When the sharing of electrons is unequal, a bond is called polar.

A special case is hydrogen, which is particularly effective in also creating bonds between different molecules, hydrogen bonds, when polarized.

Hydrogen bonds give water unusual properties that are critical for life on earth.

5.11.4 Promotion and hybridization

While valence bond theory managed to explain a number of chemical bonds so far, two important additional ingredients need to be added. Otherwise it will not at all be able to explain organic chemistry, the chemistry of carbon critical to life.

Carbon has two unpaired 2p electrons just like oxygen does; the difference between the atoms is that oxygen has in addition two paired 2p electrons. With two unpaired electrons, it might seem that carbon should form two bonds like oxygen.

But that is not what happens; normally carbon forms four bonds instead of two. In chemical bonds, one of carbon's paired 2s electrons moves to the empty 2p state, leaving carbon with four unpaired electrons. It is said that the 2s electron is “promoted” to the 2p state. This requires energy, but the energy gained by having four bonds more than makes up for it.

Promotion explains why a molecule such as CH$_4$ forms. Including the 4 shared hydrogen electrons, the carbon atom has 8 electrons in its outer shell, so its shell is full. It has made as many bonds as it can support.

However, promotion is still not enough to explain the molecule. If the CH$_4$ molecule was merely a matter of promoting one of the 2s electrons into the vacant 2p$_y$ state, the molecule should have three hydrogen atoms under 90 degrees, sharing the 2p$_x$, 2p$_y$ and 2p$_z$ electrons respectively, and one hydrogen atom elsewhere, sharing the remaining 2s electron. In reality, the CH$_4$ molecule is shaped like a regular tetrahedron, with angles of 109.5 degrees between all four hydrogens.

The explanation is that, rather than using the 2p$_x$, 2p$_y$, 2p$_z$, and 2s states directly, the carbon atom forms new combinations of the four called “hybrid” states. (This is not unlike how the torus-shaped $\psi_{211}$ and $\psi_{21-1}$ states were recombined in chapter 4.3 to produce the equivalent 2p$_x$ and 2p$_y$ pointer states.)

In case of CH$_4$, the carbon converts the 2s, 2p$_x$, 2p$_y$, and 2p$_z$ states into four new states. These are called sp$\POW9,{3}$ states, since they are formed from one s and three p states. They are given by:

$$\renewedcommand{arraystretch}{1.8} \begin{array}{l} \v... ...vert\mbox{2p}_y\rangle+\vert\mbox{2p}_z\rangle) \end{array}$$

where the kets denote the wave functions of the indicated states.

All four sp$\POW9,{3}$ hybrids have the same shape, shown in figure 5.12.

Figure 5.12: Shape of an sp$\POW9,{3}$ hybrid state.

The asymmetrical shape can increase the overlap between the wave functions in the bond. The four sp$\POW9,{3}$ hybrids are under equal 109.5 degrees angles from each other, producing the tetrahedral structure of the CH$_4$ molecule. And of diamond, for that matter. With the atoms bound together in all spatial directions, diamond is an extremely hard material.

But carbon is a very versatile atom. In graphite, and carbon nanotubes, carbon atoms arrange themselves in layers instead of three dimensional structures. Carbon achieves this trick by leaving the 2p-state in the direction normal to the plane, call it p$_x$, out of the hybridization. The two 2p states in the plane plus the 2s state can then be combined into three sp$\POW9,{2}$ states:

$$\renewedcommand{arraystretch}{2.3} \begin{array}{l} \v... ...aystyle\frac{1}{\sqrt2}}\vert\mbox{2p}_y\rangle \end{array}$$

Each is shaped as shown in figure 5.13.

Figure 5.13: Shapes of the sp$\POW9,{2}$ (left) and sp (right) hybrids.

These planar hybrids are under 120 degree angles from each other, giving graphite its hexagonal structure. The left-out p electrons normal to the plane can form pi bonds with each other. A planar molecule formed using sp$\POW9,{2}$ hybridization is ethylene (C$_2$H$_4$); it has all six nuclei in the same plane. The pi bond normal to the plane prevents out-of-plane rotation of the nuclei around the line connecting the carbons, keeping the plane rigid.

Finally, carbon can combine the 2s state with a single 2p state to form two sp hybrids under 180 degrees from each other:

$$\renewedcommand{arraystretch}{2.3} \begin{array}{l} \v... ...(\vert\mbox{2s}\rangle-\vert\mbox{2p}_z\rangle) \end{array}$$

An example sp hybridization is acetylene (C$_2$H$_2$), which has all its four nuclei on a single line.

Key Points

The chemistry of carbon is critical for life as we know it.

It involves two additional ideas; one is promotion, where carbon kicks one of its 2s electrons into a 2p state. This gives carbon one 2s and three 2p electrons.

The second idea is hybridization, where carbon combines these four states in creative new combinations called hybrids.

In sp$\POW9,{3}$ hybridization, carbon creates four hybrids in a regular tetrahedron combination.

In sp$\POW9,{2}$ hybridization, carbon creates three hybrids in a plane, spaced at 120 degree intervals. That leaves a conventional 2p state in the direction normal to the plane.

In sp hybridization, carbon creates two hybrids along a line, pointing in opposite directions. That leaves two conventional 2p states normal to the line of the hybrids and to each other.

5.11.5 Ionic bonds

Ionic bonds are the extreme polar bonds; they occur if there is a big difference between the electronegativities of the atoms involved.

An example is kitchen salt, NaCl. The sodium atom has only one electron in its outer shell, a loosely bound 3s one. The chlorine has seven electrons in its outer shell and needs only one more to fill it. When the two react, the chlorine does not just share the lone electron of the sodium atom, it simply takes it away. It makes the chlorine a negatively charged ion. Similarly, it leaves the sodium as a positively charged ion.

The charged ions are bound together by electrostatic forces. Since these forces act in all directions, each ion does not just attract the opposite ion it exchanged the electron with, but all surrounding opposite ions. And since in salt each sodium ion is surrounded by six chlorine ions and vice versa, the number of bonds that exists is large.

Since so many bonds must be broken to take a ionic substance apart, their properties are quite different from covalently bounded substances. For example, salt is a solid with a high melting point, while the covalently bounded Cl$_2$ chlorine molecule is normally a gas, since the bonds between different molecules are weak. Indeed, the covalently bound hydrogen molecule that has been discussed much in this chapter remains a gas until especially low cryogenic temperatures.

Chapter 10.2 will give a more quantitative discussion of ionic molecules and solids.

Key Points

When a bond is so polar that practically speaking one atom takes the electron away from the other, the bond is called ionic.

Ionic substances like salt tend to form strong solids, unlike typical purely covalently bound molecules like hydrogen that tend to form gases.

5.11.6 Limitations of valence bond theory

Valence bond theory does a terrific job of describing chemical bonds, producing a lot of essentially correct, and very nontrivial predictions, but it does have limitations.

One place it fails is for the O$_2$ oxygen molecule. In the molecule, the atoms share their unpaired 2p$_x$ and 2p$_z$ electrons. With all electrons symmetrically paired in the spatial states, the electrons should all be in singlet spin states having no net spin. However, it turns out that oxygen is strongly paramagnetic, indicating that there is in fact net spin. The problem in valence bond theory that causes this error is that it ignores the already paired-up electrons in the 2p$_y$ states. In the molecule, the filled 2p$_y$ states of the atoms are next to each other and they do interact. In particular, one of the total of four 2p$_y$ electrons jumps over to the 2p$_x$ states, where it only experiences repulsion by two other electrons instead of by three. The spatial state of the electron that jumps over is no longer equal to that of its twin, allowing them to have equal instead of opposite spin.

Valence bond theory also has problems with single-electron bonds such as the hydrogen molecular ion, or with benzene, in which the carbon atoms are held together with what is essentially 1.5 bonds, or rather, bonds shared as in a two state system. Excited states produce major difficulties. Various fixes and improved theories exist.

Key Points

Valence bond theory is extremely useful. It is conceptually simple and explains much of the most important chemical bonds.

However, it does have definite limitations: some types of bonds are not correctly or not at all described by it.

Little in life is ideal, isn’t it?